Figurate Numbers Calculator (Triangular, Square, Pentagonal, Hexagonal)
Calculate triangular, square, pentagonal, and hexagonal numbers from the 1st to the Nth term and compare them on a graph. Instantly compute the value of any nth term. Includes a reference table for terms 1-15.
Reference table for terms 1-15 (triangular, square, pentagonal, hexagonal)
A table listing the values of the four types of figurate numbers from the 1st to the 15th term.
| n | Triangular numbers | Square numbers | Pentagonal numbers | Hexagonal numbers |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 |
| 2 | 3 | 4 | 5 | 6 |
| 3 | 6 | 9 | 12 | 15 |
| 4 | 10 | 16 | 22 | 28 |
| 5 | 15 | 25 | 35 | 45 |
| 6 | 21 | 36 | 51 | 66 |
| 7 | 28 | 49 | 70 | 91 |
| 8 | 36 | 64 | 92 | 120 |
| 9 | 45 | 81 | 117 | 153 |
| 10 | 55 | 100 | 145 | 190 |
| 11 | 66 | 121 | 176 | 231 |
| 12 | 78 | 144 | 210 | 276 |
| 13 | 91 | 169 | 247 | 325 |
| 14 | 105 | 196 | 287 | 378 |
| 15 | 120 | 225 | 330 | 435 |
What figurate numbers are
A figurate number counts the dots when you arrange them regularly into a polygon. Arranged in a triangle you get the triangular numbers 1, 3, 6, 10 and so on; in a square, the square numbers 1, 4, 9, 16; in a pentagon, the pentagonal numbers 1, 5, 12, 22. **The square numbers coincide with the perfect squares** because dots laid out in a square come to exactly n × n — a plain example of geometry and algebra meeting directly.
Each family has a tidy closed form: n(n+1)/2 for triangular, n² for square, n(3n−1)/2 for pentagonal and n(2n−1) for hexagonal. Relationships between them are known too — **two consecutive triangular numbers sum to a square number**, and **the hexagonal numbers coincide with the odd-indexed triangular numbers** — and watching those unexpected connections emerge is much of the appeal. This tool computes from the first term to the Nth and compares them on a graph, and can also give you a single term directly.
How to compute figurate numbers
- Choose the family Select triangular, square, pentagonal or hexagonal.
- Set how many terms to show The first through the Nth term appear as a list and a graph.
- Compare the growth on the graph It becomes visible that the more sides the polygon has, the faster the sequence climbs.
- Obtain a single term For a large n, you can ask for the value of that term alone.
Tips for getting more out of it
- Triangular numbers T(n) = n(n+1)/2 count the total number of points when stacked into an equilateral triangle. The classic bowling pin arrangement (10 pins) is triangular number T(4) = 10.
- Square numbers S(n) = n² count the total number of points arranged in a square. They are also known for the property that summing consecutive odd numbers produces a square number (1, 1+3=4, 1+3+5=9, ...).
- Pentagonal numbers P(n) = n(3n-1)/2 and hexagonal numbers H(n) = n(2n-1) count points arranged into a regular pentagon and hexagon respectively. Every hexagonal number is also a triangular number (H(n) = T(2n-1)).
- The graph lets you compare how fast each of the four types grows. For the same n, shapes with more vertices (3, 4, 5, 6) tend to produce larger values.
- This tool supports calculations up to the 1000th term. Since values grow rapidly for larger n, the graph and table display are limited to 1-30 terms for readability.
Where this helps
Studying sequences
Match the derivation of a closed form against the actual values to confirm what the formula means.
Verifying relationships between sequences
Check with real figures that two consecutive triangular numbers really do sum to a square.
Programming exercises
Compare a function you wrote to produce figurate numbers against the correct values.
Background for puzzles and competition mathematics
Figurate numbers turn up often in number theory problems, and a table of values makes it easier to spot the pattern.
Figurate number terms explained
- Triangular number
- The count of dots arranged in a triangle, with closed form n(n+1)/2. **It equals the sum of the integers from one to n.**
- Square number
- The count of dots arranged in a square, namely n².
- Pentagonal number
- The count of dots arranged in a regular pentagon, namely n(3n−1)/2.
- Hexagonal number
- The count of dots arranged in a regular hexagon, namely n(2n−1). **These coincide with the odd-indexed triangular numbers.**
- Closed form
- An expression giving the nth term of a sequence in terms of n. Every family of figurate numbers has a tidy one.
- Polygonal number theorem
- The result that every natural number is the sum of at most three triangular numbers, four squares, or five pentagonal numbers.
Frequently Asked Questions
Side Note — The Pythagoreans and the "mystery of numbers" found in figurate numbers
The study of figurate (polygonal) numbers dates back to the Pythagorean school of ancient Greece around the 6th century BCE. Guided by the belief that "all is number," they arranged pebbles into geometric shapes to visually understand the properties of numbers. The very names "triangular" and "square" numbers come from the shapes formed when the points are laid out.
There is a beautiful relationship between triangular and square numbers: adding any two consecutive triangular numbers always produces a square number (e.g., T(3)+T(4) = 6+10 = 16 = 4²). This becomes intuitively clear when you actually draw the arrangement of points. Geometric proof techniques like this had a major influence on the later development of number theory.
Figurate numbers remain an active subject of study in modern mathematics — for instance, the property that "every hexagonal number is also triangular," or the search for numbers that satisfy multiple figurate-number definitions at once, are still popular exercises in number theory today. It is remarkable that many properties the Pythagoreans discovered 2,500 years ago, arranging pebbles by hand with no paper or written arithmetic, still hold as valid theorems today.